Volume of a Right Circular ConeDate: 10/07/2001 at 19:45:43 From: Jeffrey A Dozier Subject: Volume of a right circular cone Using calculus, derive the formula for the volume of a right circular cone with a radius of r and height h. I seem to remember that there is a way using calculus to create a proof showing that the volume of a right circular cone is: pi r^2 h/3 . Date: 10/07/2001 at 23:11:54 From: Doctor Paul Subject: Re: Volume of a right circular cone Draw an x-y coordinate system. In the first coordinate, draw a line from the origin to some point (h,r). This represents the line y = r/h * x. We want to rotate this line about the x axis and compute the volume of the solid obtained thereby. Note that the resulting solid would be a right circular cone with height h and radius r. The formula for the volume of a solid of revolution is given by: Pi * integral of f(x)^2 dx so what we have is: Pi * integral from 0 to h of ((r/h)*x)^2) dx square the integrand and pull out the constant r^2/h^2 then we have: Pi*r^2/h^2 * integral from 0 to h of x^2 dx the integral becomes 1/3 * h^3 when evaluted. When 1/3 * h^3 is multiplied by the constants that were outside the integral, we get the desired answer: 1/3 * Pi * r^2 * h I have been deliberately brief in the hope that you can fill in any missing details. If something is unclear, please write back. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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